See Section REF  for more precise definition. The problem of this random extension is that the transformation \(\widetilde{T}\)  is no longer invertible, therefore results like Shannon-McMillan-Breiman theorem for conditional measures (Proposition REF ) cannot be directly used here, hence the cascade version of the fact that \(\Phi (\widetilde{\theta }^+_{t,\mathbf {i}^-,\mathbf {j}^-})\)  and \(\Phi (\widetilde{\mu }^+\times \widetilde{\nu }^+)\)  have the same dimension, or even the simpler statement for \(\Phi _{\mathbf {I}}(\widetilde{\mu }^+_{\mathbf {i}^-})\)  and \(\Phi _{\mathbf {I}}(\widetilde{\mu }^+)\)  as in Theorem REF , becomes much harder to prove. We will bypass this difficulty by combining the methods developed in [1]}, [2]}, [3]}.
